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Masters Theses Student Theses and Dissertations

1965

Effects of internal pressure upon the buckling of thin circular Effects of internal pressure upon the buckling of thin circular

cylindrical shells under axial compression cylindrical shells under axial compression

LeRoy E. Thompson

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Recommended Citation Recommended Citation Thompson, LeRoy E., "Effects of internal pressure upon the buckling of thin circular cylindrical shells under axial compression" (1965). Masters Theses. 5231. https://scholarsmine.mst.edu/masters_theses/5231

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EFFECTS OF INTERNAL PRESSURE UPON THE BUCKLING OF THIN CIRCULAR CYLINDRICAL

SHELLS UNDER AXIAL CO?-'I.I:'RESSI ON

BY ·J' ·.

LEROY E: THOMPSON

A

THESIS

submitted to the faculty or the

UNIVERSITY OF MISSOURI AT BOLLA

1n partial fulfillment of the requirements for the

MASTER OF SCIENCE IN CIVIL ENGINEERING

Rolla, Missouri

1965

. -7 (,'

I I ~ (/

ABSTRACT

An experimental inve.stip-ation was made of the

effects of internal pressure upon the buckling of thin

circular cylindrical shells under axial compression.

ii

A comparison was made of the experimental and theoretical

results; the latter were derived from the lar~e-deflection

theory. Results showed that the internal pressure had an

appreciable strenptheninp: effect upon the bucklin~ of the

cylindrical shell. The comparison showed that the

experiment;al results were lower than the theoretical

results for a given internal pressure. The discrepancy

between the results can be expected since there were

factors not included in the theory: imperfections, end

conditions, lengths, and material irregularities.

iii

ACKNOWLEDGEMENT

The author wishes to express his ap~reciation to

Dr. Joseph H. Senne, Jr., Professor of Civil Engineerin~,

University of Missouri at Rolla, who gave constructive

su~~estions and guidance on the preparation of this

paper. He would also like to thank Professor R. F.

Davidson, Chairman of the Department of Mechanics, for

his permission to use the facilities of its laboratory.

The author also wishes to thank J. E. Spooner, Assistant

Professor of Civil Engineerinp., who read the paper and

made helpful supgestions.

iv

TABLE OF CONTENTS

Page

LIST OF FIGURES • . . • . LIST OF TABLES. NOMENCLATURE. • • • . . .

. . . . . . . . . . . . . . . . . . . • v

. • vi .vii

I.

II.

III.

IV •

v. VI.

INTRODUCTION .. . . . . . . . . . . . . . 1

REVIEW OF LITERATURE. . . . . . . . THEORY. • • . . . . . . . . . . . . . . . . . .

4

9

DISCUSSION. . . . . . . . . . . . • 16

A. Description of Apparatus. . • . .... 16 1. Test Specimens. . • . • . . • . .. 16 2. Material Specimens. • • . • . . . . . • 20 3. Instrumentation . • • . . • 22 4. Equipment • . • • . • . . . . . 25

B. Experimental Procedur·es • • . . . • • . 25 C. Experimental Results. . • . . • . . 21

CONCLUSIONS . • • • . • . . . . . . . . . . . . 34

RECOMMENDATIONS • . . . . . . • . . . . . . 36

BIBLIOGRAPHY. • . . . . . . . . • . . . . . . . . . . 37 VITA. • • • . . . • . . . . . . . . . . . . . 40

v

LIST OF FIGURES

Fipure Page

1. Bodies of the cylindrical shell specimens. . 17

2. Bottom and top ends of specimen A. • . . . 18

3.

4. 5.

Bottom end of specimen B • . . . . 21

Top end of specimen B. . . . . . . . . . 21

Strain gape positions. . . . . . . . . 23

6. Switching units and strain indicators •.•..• 24

1.

8.

9.

Testing setup. . • • • • . . • • • • • • • • • •

Experimental results of the buckling stress at various pressures . • • . . • . • .•

Experimental values of the buckling load at various internal pressures • . • . . .

• 26

29

• 30

10. Post-buckling failure of specimen A. • • . . . . 33

11. Comparison or theoretical and experimental results showing the increment of bucklin~ stress due to internal pres sure • • . • . . • • . . • • . . • • . . 3.5

vi

LIST OF TABLES

Table Pa~e

I. Theoretical buckling stresses ~or various internal pressures. . . • • . . . . . . .15

II. Experimental buckling stresses for various internal pressures. . . . . . . • . . . .28

Symbols:

fo, :rl' r2

m

n

p

t

NONENCLATURE

parameters used in the deflection function

vii

number or waves in longitudinal direction within length equal to circumference of cylindrical shell

number or waves in circumferential direction

internal pressure, psig

thickness of cylindrical shell wall, inches

average compressive stress, psi

aspect ratio or buckled waves

Poisson's ratio

Young's modulus, psi.

radius of cylindrical shell, inches

functions of {3

runctions of p and ~

(D represents the functions D1 , n2 , .. D5

)

(D represents the functions D1 , n2 , •• D5) length of cylindrical shell, inches

Nondimensional parameters:

~:§:R E t

viii

.6.Q-cr = Gcr - <~cr) p : o

p = ~(~t ~~ : (~- ~)?Dl

{J Q = work or enerpy parameter

Subscripts:

u

cr

unbuckled state just prior to buckling

buckling condition

I

INTRODUCTION

1

The reason for conducting research in this particular

area is to study the increased efficiency obtained from a

thin circular cylindrical shell under axial load by

including an additional loading of internal pressure.

The structure is efficient if its material is used to the

maximum limit of its permissible working stresses. When

a minimum of dead wei~ht of the structure is the critical

design condition, the most structurally efficient design

will in all probability yield the most economical

structure. In modern spacecraft, aircraft, and missiles,

each pound of dead weight saved will yield many other

additional advantapes. One such advantage is that an

additional .tJOund may be included into the payload of the

particular vehicle. This ~ayload may be either cargo,

warhead, test equipment, or surveillance systems. Another

advantape is that the range of the vehicle can be

increased when the weipht saved may be used for additional

propulsion fuel. As is already known, the dead weight can

be minimized by selecting a workable material with low

specific weight and yet the highest permissible working

stresses for the given unit wei~ht; but this same

objective can be obtained also by the extensive use of

pressurized cylindrical shell structures. Many times,

pressurization may be dictated by circumstances other than

for ~tructural purposes, such as the case where internal

systems must be pressurized. Also, if the structure is

sealed and is launched into a less dense atmosphere or

into space, the external pressure will decrease while

the pressure internally is maintained near the initial

condition at the time of launching; consequently this

will yield the internal pressure loading upon the

structure. Therefore, even thou~h some shells may not

2

be pressurized for structural purposes the structure may

have this type of loading already, and it can be utilized

in addition for a structural desip-n use. This internal

pressure will materially increase the stress at which

buckling will occur in a circular cylindrical shell.

The increase in the buckling stress then definitely

achieves a higher structural efficiency and thus a

considerable weipht saving.

Many experiments have been conducted (3,5,11,12,13,

24.,25) on thin pressurized circular cylindrical shells

under axial compressive load, but the results are far

from being complete. It has been found that the

theoretical investigations cannot be relied u~on for

design purposes due to the discrepancies found between

experimental and theoretical buckling loads. The reason

for these discrepancies is that the deflection theory

which most accurately describes the ~roblem does not

account for all the exact numerical values of the

parametric coefficients which experimentation have found

to be critical. In this paper, the experimentation will

be extended to find some of the numerical coefficients

or shells with diff'erent map:nitudes or parameters and

the results compared to existing theory.

3

4

II

REVIEW OF LITERATURE

The first theoretical investigation of the buckling

of thin-walled cylindrical shells under axial compression

and lateral pressure was done by Flugge (23) in 1932.

r,lugge 's conclusion v.ras that the internal pressure effects

on the bucklinp load were negligible. This conclusion was

contradictory to the experimental and theoretical work

which followed in the years thereafter. This early

theoretical analysis dif.fers because Flugge' s analysis '\<:as

based upon the small, rather than the lar~e, deflection

theory. The small-de.flection theory was f'ound to be

applicable only when the deflections are small with

respect to the wall thickness.

Since cylindrical shells can have very lar~e

deflections without reaching yield stress, the

lar~e-deflection theory is the most accurate theoretical

approach. This theory neglects the local bending stresses

with respect to the membrane stresses. It was assumed

that the external forces, uniformly distributed along the

edge o.f the shell, were tangent to the meridians (1,2,4,

6, 7).

The use of larpe-deflection theory for shells under

axial compression was First advanced by Von Karman and

Tsien (6) in an attempt to explain the discrepancies

between the bucklinp loads predicted by theory and those

5

obtained from experimental results. The results indicated

that cylindrical shells can be maintained in equilibrium

in the buckled state by a compressive load considerably

lower than that predicted by theory. Thererore, the

cylindrical shells desiFned by the theoretical method

might possibly fail. The treatment of Von Karman and

Tsien was found later to be incomplete since the total

potential energy was not differentiated with respect to

all the physical parameters involved.

Donnel1 (10) first introduced a set of simpliryinp

assumptions which are now commonly used. From these

assumptions Donnell initated the ~eneralized equations

for the new large-deflection theory.

Based upon the observed contradictions of

experimental results with Flu~ge's conclusion, Lo, Crate,

and Schwartz (11) devised a revised larpe-deflection

theory~ using the method of Von Karman and Tsien. The

results showed that the buckling load increases with

increasinp values of internal pressure up to a limit

where it remains nearly constant. The large-deflection

theoretical buckling loads for the lar~e values of

internal pressures were then comparable to the

small-deflection theory.

The tests by Dow and Peterson (26) showed bucklin~

~oads that were considerably above the average from other

experiments. This was at least partly due to the fact

that their test cylinders were pressurized by oil, so

that buckling was accompanied by a rise in internal

pressure. Another factor was round to be non-uniformity

of loading around the shell circumference which caused a

lar~e amount or scattered data in the test results.

The photoelaetic experimental work conducted by

Tennyson (22} indicated that buckling loads were within

ten percent or the classically predicted values and all

the tested shells behaved completely elastically, thus

permittinp repeatable tests.

Harris, Suer, Skene, and Benjamin (13) developed a

semi-empirical procedure which permitted an axial

compressive loading and internal pressure buckling

analysis of cylindrical shells with a knowledge of the

cylinder geometry only. This analysis was achieved by

correlating experimental data statistically with

theoretical parameters. Fung and Sechler (12) proposed

6

a design method which gave slightly conservative values

for the buckling stress or axial compression and internal

pressure loadings on a cylindrical shell.

Sergev (15,16) and Walton (15) conducted experimental

tests on the strength or pressurized columns. The

research indicated that the instability criterion was

entirely dirrerent for the two cases of pressurized

columns with open and with closed ends. The results also

indicated that nothin~ was ~ained in the critical buckling

load by pressurizing the columns; although the stress

state in the column wall was very dirrerent from the

7

unpressurized cnse. This was due to the fact that their

columns had lar~e slenoerness ratios and small radius to

thickness ratios.

Almroth and Brush ( 24) worked with the upper and

lower bounds of a bucklinp- load which were not sensitive

to chance variation for pressurized cylinders and for

cylinders filled with a soft elastic core. It was found

that these bounds conver~ed with increasinfl internal

pressure or core stiffness.

The results of the analysis by Siede (25} indicated

that shearin~ stresses between the elastic core and the

cylindrical shell were assumed to be negligible, so that

restraint was offered only ap-ainst normal displacements.

A closed-form solution for the elastic support offered

by the core was piven and comparisons were made with

experimental data.

Donnell and Wan (9) studied theoretically the effects

of the initial imperfections of a cylindrical shell

upon the critical bucklinp loads. It was found ths_t the

imperfections caused the buckling to occur prematurely.

This remains a purely static theory containing idealized

assumptions which cannot be verfied practically.

Lee (18) made both an analytical and experiment~

study of inelastic instability of initially imperfect

cylindrical shells subjected to axial compression. The

comparisons of experimental with theoretical results

indicated that the application of the deformation theory

provided a rairly accurate prediction or buckling

strenpth, but railed to yield a correct description or

the post-buckling behavior. The buckling strenpth was

over-estimated by using an incremental theory with the

initial imperrections being considered.

8

Almroth, Holmes, and Brush (29) concluded that

minimization or initial imperrections in axially

compressed cylindrical shells preatly increased the

buckling load, and the magnitude or the minimum

post-buckling equilibrium load was relatively insensitive

to initial imperrections.

It has been round that many ractors combine to

determine the buckling load or a particular cylinder.

These include care in fabrication, experience or the

investigator, end conditions or the test specimen,

initial derormation, limits or strain regions, runctions

or dirrerent cylindrical shell parameters, etc. The

nature or the shell problem seems to be that design values

ror buckling load and internal pressure must be round

by experimental methods since current theories are not

surricient to determine exact numerical values or the

parametric coerricients.

III

THEORY

The existinP- procedure ror the computation or the

9

buckling stress by lar~e-de:flection theory, which involves

the solution o:f :four s irm1l taneous nonlinear equations :for

each pressure loadinp, was advanced by Von Karman and

Tsien (8). Improvements in these rour equations were made

by other investi~ators in which the potential energy was

properly dirferentiated with respect to all physical

parameters and the effects or internal pressure (11) were

included.

The solution or the four simultaneous equations shown

below (Eqs. 1, 2, 3, and 4) required a tedious and lenpthy

numerical process. Lo, Crate, and Schwartz (11) improved

this numerical process by the introduction of one more

equation (Eq. 5) which derined the condition at buckling.

These rive simultaneous equations enabled the solution to

be found for the particular condition at buckling. Where

previously four simultaneous equations had to be solved

several times in order to get sufricient results to plot a

stress-strain curve relationship in order that the

critical condition at buckling could be detected.

i'' = (?S)2 (2D2 ) (?S)D3 + 2 72n

5 (Eq. 1)

i'' (?.,S') 2(2D2 ) (?S)(l.5D3 ) + n

4 + YJ

2D5

(Eq. 2)

i'' - [<?5)2 (D2) - (75)(D)) + (DIJ) r r 2 r- J n1

+ ? <ns>ro (Dl >F (Eq. 3 >

~· - E>JS")2(D2~ - ("? 5) (D3 ~ + (D4 ~

2 J -+ -ry (D5) .!. - L"l')D y9 2 1>2 1.

-, () = (~$)2(D2 - ~12) - (?~)D3 + D4 + ?205

The nomenclature or the terma are as rollows:

- G"'R (} = Et

~ = average compressive stress

E = Youn~' s modulus

R = radius or cylindrical shell

t = thicknees or cylindrical shell wall

p = internal pressure

~ = ~ = aapect ratio or buckled waves

10

(Eq. 4)

(Eq. 5)

m = number or waves in longitudinal direction within

length equal to circumrerence or cylindrical

shell

n - number or waves in circumrerential direction

'?- n2j

D1 = !l(~ + F + P2)

r2 F=r

1

~0' rl, ~2 =parameters used in deflection function

R 5 = t'1t

D2 = a1

+ B2

f> + B p 2 + B ~3 + !a 'f>4 3 4 2 4

D3 = (2s4 + /rr) (1 + 2'f)

D4 = (2B4 + -ft)+ !<F+ F2)

2 D5 = B5 + B6 (f> + 'P )

11

1 fi + ,.s4 ll f£4 -134 4 ::t ~ = b'Q:[ 8 + 4 (1 +~2 )2 + (1 + 9~2 > 2 + t91jB2 ) j

- 1 [.2_ .,84 .,e4 ~h. ~ B2 - YO 2 ( l + )€ 2 ) 2 + ( 1 + 9{32) 2 + ( 9 + .,82) ~

1 [11 .,e4 .s4 .).t .... ;] B3 = !bL2 (1 +-82)2 + (1 + 9~2)2 + ~>]

l. -/34 B4 = 'S" (1 + ·{!?>2

1 11 2 2 l. l+J Bs = ,.....6 c-t__,;-~f-·2 > ~ c 1 + 13 > + q: < 1. + ,B j

B6 = 6(1 ~ p 2 ) (1. +#-> fA = Poi~~on' s ratio

2 (Dl )F = ~,B (1 + 2P)

(D2 )~ = B2 + 2B3p + 3B4~

2 + 2B4p3

(D))f> = 2(B4 + ~)

(D4 )~ = !<1 + 2F)

(DS }f = B6 (1 + 2'P)

(D1 ).,g = 2D1

( D2 ~ - Bl • + B2 'f' + B3 I p2 + B4. f>3 + ~.1\. ~4

(D)~ - 2B4 ' (1 + 2p)

- 2B4

•

- B5' + B6' ( F + F2)

12

B' 1 -.ft;Et + ~ ~>3 + (1 f9{fl 3 + 9 (9 #.,s2 >~ - ft[~ (1 !~2 )3 + (1 t\/>3 + 9(9 .tl'~ll~

The four equations (Eqs. 1~ 2, 3, and 4) were derived

in the following manner. The expressions for the ener~y

due to elastic extensional~ bending, applied compressive

load, and the internal pressure were found. The summation

of these four energy expressions was the equation ror the

total potential energy. The total potential enerp,y

equation was differentiated with respect to the initial

deflection function parameter (t0

) and the derivative was

eet equal to zero. The resulting expression was

substituted into the four energy expressions, which in

turn were expressed in terms of the nondimensional

13

parameters ~, p, f, 1' $, and W. The term W is the

nondimensional parameter ror energy. The summation of

the rour nondimensional parameters of energy formed the

nondimensional total potential energy parameter in the

buckled state. The equilibrium positions of the

cylindrical shell in the buckled state were obtained by

differentiating the nondimensional total potential energy

with respect to each parameter ,, ~, f, and ~ and by

setting the derivatives equal to zero. The results were

~our simultaneous nonlinear equations. The D and B terms

~or the certain functions of ~ and F as well as the

nondimensional parameter of average compressive stress f' were then substituted into the above four simultaneous

nonlinear equations, which yielded Eqs. 1, 2, 3, and 4. Equation 5 was derived in the ~ollowing manner. An

expression was written for the work done by the pressure

during buckling. The strain energy expression for the

buckled state was written as the sum of the strain energy

in the unbuckled state just prior to buckling and the work

done by the pressure durin~ buckling. A relationship of

end shortening and avera~e compressive stress was written

for the buckled and unbuckled states. The fact was then

reco~nized that the end shortenin~ remains unchanged

during buckling from the unbuckled state to the buckled

state, provided the loadin~ machine is assumed to be rigi~

Therefore, the relationships for the buckled and unbuckled

states could be equated and a relationship among

lh

the nondimensional parameters or average compressive

stresses could then be determined. This relation as well

as the nondimensional parameter or averape compressive

stress ~~ was then substituted into the strain enerpy

expression. The nonlinear equation (Eq • .5} was thus

obtained for the bucklin~ criterion.

The solution or the rive equations (Eqs. 1, 2, 3, 4,

and 5) pives the buckling stress ror a given internal

pressure. The method for the solution or these five

nonlinear simultaneous eauations was computed in the

rollowinp manner

thus determining

( 11}.

r;s. Equations 2 and 5 were equated,

2 In like manner, ? was obtained

using Equations 1 and 2. For a preassigned value of ~~

various values of Fwere assumed. Then ?S ano ?2 were

computed, which were substituted into Equations 1 and 3

to obtain the ~· for each of the two equations,

respectively. The resulting two ~' values were plotted

against the assumed various values of f. The pair of

curves would intersect at a common value of both f' and

r· The corresponding values of ?s and ?2 were computed

and substituted in Equation 4 so the nondimensional

parameter of pressure p could be determined. For each

assipned value or~, the corresponding values of ?• and p

were thus obtained. The relationship between ~· and ~ was

known, so the corresponding values of ~ and p were thus

obtained (Table I). When the factor?!= 0, the classical

15

critical bucklinp stress o:f 0.605 occurred as tbe cut-off

buckling stress which was independent of pressure (11).

Table I * Theoretical buckling stresses ror

various internal pressures

- - ~Cf'cr p <r'

0.000 0.376 o.ooo

0.020 0.444 o.o68

0.040 0.4AO 0.104

0.060 0.506 0.130

0.080 0.528 0.152

0.100 0.547 0.171

0.120 0.565 0.189

0.140 0.581 0.205

0.168 0.605 0.229

*Reference (11), pape 23.

IV

DISCUSSION

A. Description of Apparatus.

1. Test Specimens.

16

The two specimens used for the experimental

tests are hereby described as specimen A and

specimen B. Specimen B was used for the internal

pressure-bending moment interaction tests conducted

in reference (27).

Test Specimen A: The specimen used for the

tests was a cylindrical shell, 30 inches lonp with a

15 inch diameter, made of No. 28 u.s. standard gape

galvanized sheet steel. This ga~e is equivalent

to a nominal thickness of 0.021 inch.

The butt-joint of the two longitudinal edges

was covered by inside and outside splice straps of

the dimensions 0.021 inch thick and 1! inches wide.

The straps were riveted with 1/8 inch diameter

rivets spaced in a staggered arrangement 3/8 inch

on center alonp, the total length of the cylinder

(Fig. 1). Soldering was done around the rivet heads

and edges of straps in order to properly seal the

butt-joint.

Both the top and the bottom ends {Fi~. 2) were

made of an 18 inch square by 3/8 inch thick plywood

with a mounted 15 inch diameter disk of 3/4 inch

thick plywood to act as an inside shoulder support

x 0.021" splice strap ~~~--(inside and outside, specimen A)

(lapped, specimen B)

No. 28 gage sheet steel

1/8" diam. rivets t

No. 28 gage sheet steel

0 0

typ

30" specimen A 27" specimen B

--if---+- ------t--

1 t=0.021"

Fig. 1. Bodies or the cylindrical shell specimens.

17

18

Fig. 2. Bottom and top ends or specimen A.

19

ror the shell. There was a piece or rubber, 18

inches square by 1/8 inch thick, placed between the

two pieces or plywood. The plywood ends were then

bolted to an 18 inch square by 3/4 inch thick A-7

steel plates. Steel balls were tack-welded to the

steel plates in order to apply a concentric

concentrated compression load. Strain ga~e wire

outlets and the air pressure inlet were provided on

the top end. They were sealed with an epoxy cement

or the compound Epibond 104 with its hardener 951.

The plywood ends were also sealed to the shell by

the epoxy cement arter the disks were rorce ritted

within the inside shell diameter.

Test Specimen B: The specimen used :for the

tests was a cylindrical shell, 27 inches long with

a 15 inch diameter, made or No. 28 U.S. standard

gape galvanized sheet steel. The gage is equivalent

to a nominal thickness of' 0.021 inch.

The butt-joint or the two lonp:itudinal edges

was covered by a lapped splice strip with the

dimensions of' 0.021 inch thick and 1-i inches wide.

The splice was riveted with 1/8 inch diameter

rivets spaced in a stapgered arranFement 3/8 inch

on center alonp the total length or the

cylinder (Fig. 1}. Soldering was done around the

rivet heads and edges or splice in order to properly

seal the butt-joint.

20

Both the top and the bottom ends (Fip-s. 3, 4)

were made of t inch A-7 steel plates and then welded

to 15 inch outside diameter steel rings l inch thick

and 1 inch in hei~ht. The rings acted as an inside

shoulder support ror the shell. Strain page wire

outlets and the air pressure inlet were provided

on the top end. They were sealed by Smooth-on No. 1

iron cement. The joints between the cylindrical

shell and ends were arc-welded using nickel rods.

The quality or welding was not good due mainly to

the dirrerence in thickness or materials. As a

result there was minor leakage or air found along

these joints.

2. Material Specimens.

The modulus of elasticity, yield strength, and

ultimate strength of the galvanized sheet steel

were experimentally determined. Five strips of a

nominal width of 0.750 inch by a thickness of 0.021

inch were tested for the above mentioned properties.

The strips were instrumented with SR-4 electric

strain gages type A-1 in order to record the tensile

strains. The results were found to be 29.3 x 106

psi. for the modulus of elasticity, 42000 psi. for

the yield strenFth, and 52000 psi. for the ultimate

strenp:th.

21

r 18" square-------...- _1"=1 r-t" A-7 steel

plate

i" A-7 steel ring

Fig. 3. Bottom end or specimen B.

A-7 steel plate

1 Air inlet 2 Wire outlets 3 Air pressure gage

Pig. 4. Top end or specimen B.

22

3. Instrumentation.

The two cylindrical specimens were instrumented

with the same patt~rn o.f strain g-apes. 'fr1ere were 12

SR-4 electric strain gages equally spaced along the

outside circum.ference o.f the cylindrical shell at

mid-length, and directly opposite them on the inside

were 12 additional gages. On specimen A o.f these 24.

p.a~es, 16 were type A-1 strain gages used to measure

the strains alonp the longitudinal direction of the

shell, and the remaining 8 were AR-1 rosette strain

gages used to measure both the lonpitudinal and

circum.ferential strains (Fig. 5). On specimen B o.f

these 24 gages, 16 were type A-3 strain papes used

to measure the strains alon~ the longitudinal

direction o.f the shell, and the remaining 8 were

AR-1 rosette strain gages used to measure both the

longitudinal and circum~erential strains (Fip. 5).

The strain gapes were connected to two

switching units, a Baldwin-Lima-Hamilton 20 channel

unit and a Budd 10 channel unit. The switching

units were then connected to a Baldwin SR-4 strain

indicator and a Budd strain indicator respectively

(Fi~. 6). Th~ strains could be read to a micro-inch

per inch.

Arabic numerals denote gages measuring longitudinal strains.

Roman numeral.s denote,, gages measuring circumrerentia1 strains.

v Type A-1 strain gage, specimen A.

v Type A-3 strain gage, specimen B.

GV Type AR-1 rosette strain gage, specimens A and B.

Fig. 5. Strain gage positions.

23

24

Fig . 6 . Switchinp units and strain indicators.

25

4. Equipment.

A Tinius-Olsen 60,000 pound universal testing

machine was used. A booster-stora~e tank or 60 psi.

capacitv, manuractured by Soiltest Incorporated,

Model K-670, was used as a stand-by air source.

The tank had standard equipment or a source pressure

gage as well as an air repulator with a pressure

gage connected to the air line outlet. The

testing machine could be read within 100 pounds.

The air pressure gage could be read within one psi.

B. Experimental Procedures.

The specimens were subjected to a concentric axial

compressive load in the testing machine. Compressed air

was used to produce internal pressure, which was

maintained at any desired constant value by an air

pressure regulator. The air source with a regulator was

needed to the utmost since the leakages were appreciable

in specimen A and minor in specimen B.

The switchin~ units were adjusted so that all the

initial ga~e readings could be recorded. Then the

cylindrical shell was preloaded with an axial load slightly

greater than that which would be the total end load

created by the internal pressure. Compressed air was

next let into the cylindrical shell rrom the stora~e tank

until the desired internal pressure was obtained. The

desired internal pressure was controlled by the air

26

Fig. 7 . Testing setup .

27

rep.:ulator. rr.he axial compressive load was incr·eased in

increments until the initial buckling was observed. At

each increment or axial load all ~a~e readings were

recorded. The internal pressure was maintained at a

constant level durin~ the test procedure. The axial

compressive load was then reduced and the internal

pressure was chan~ed to another desired value. As each

value of' internal pressure was changed, the same test

procedure was repeated.

C. Experimental Results.

The experimental results are tabulated in Table II

and plotted in Fi~. 8 and Fig. 9. The Fig. 8 shows the

nondimensional critical buckling stress plotted apainst

the nondimensional internal pressure. The curves

illustrate an increase in buckling stress with an increase

or internal pressure. The rate o:r change :ror the increase

in buckling stress decreased :f'or an increase of' internal

pressure. The data :ror specimen A is not as complete as

specimen B because specimen A could not retain the high

internal presRures due to the excessive pressure losses

throv~h its ends. 'rhe term <J';:;cr is the averap-e

compressive stress when in the unbuckled state just

prior to buckling. The stress ~cr is :round by dividin~

the critical bucklinp load Per by the cross-sectional

end area of the cylindrical shell. The critical buckling

load Per occurred when the strain maintained a

constant value ror a small increase in compressive

28

Table II

Experimental buckling stresses for various internal pre~sure8

- - -p p p (JCr 6\TCr psi. lbi:

Specimen A

0 .00000 12,200 0.151 o.ooo

1 .00435 13,200 0.163 0.012

3 .01305 14,700 0.181 0.030

5 .02175 15,300 0.188 0.037

Specimen B

0 .00000 18,300 0.225 o.ooo

1 .00435 19,700 0.243 0.018

2 .00870 20,400 0.2.52 0.027

3 .01305 21,800 0.269 0.044

4 .01740 22,300 0.275 o.o5o

.5 .02175 23,000 0.283 0.058

8 .03480 24,200 0.298 0.073

12 .05220 25,500 0.314 0.089

16 .06960 27,000 0.333 0.108

20 .08700 27,800 0.343 0.118

.os ® Specimen A

0 Specimen B

Fig. 8. Experimental re~ulte or the bucklin~ etre~~ at variou~ pre~~uree.

29

({)

0... •r-1 ,!x::

30

25

.. 2 0 r--·,_,r --f.< (.)

g-.

10

5 0 Specimen A

0 Specimen B

Internal pressure "p", p~i.

Fig. 9. Experimental value~ or the buckling load at variou~ internal pre~5ure~.

30

31

load. Usually, no visible evidence or buckling could be

observed during the occurrence or this phenomenon. The

buckling occurred locally and not simultaneously at all

the strain pa~es. In Fig. 9 the critical buckling load

is plotted against the internal pressure. The curves

illustrate an increase in buckling load with an increase

or internal pressure.

The magnitudes or buckling stress and buckling load

are greater for specimen B than specimen A at a given

internal pressure. The reasons ror this di:ff'erence can

be two-:fold. The specimen A was 30 inches in length,

where the specimen B was 27 inches in length. The ends

o:r the specimen A were constructed of plywood (Fig. 2),

where the ends of the specimen B were constructed or

steel (Figs. 3 and 4). The major reason f'or the

differences in the magnitudes is believed to be due to

the construction o:r the ends. The non-uni:formity or

loading around the shell circum:rerence would be more

appreciable .for the plywood ends than :for the steel

ends. The test results indicated this in the strain

readin~s by having more variation in strains for

specimen A than specimen B. The dif'rerence in specimen

lengths is believed to be o:r minor signi:ficance, since

the change in the slenderness of the specimens was small.

Strains in the circum£erential direction o:r the

cylindrical shell increased as pressux·ization took place.

The variations or the circ~erential strains were small

32

durin~ the testing at a Fiven internal pressure.

'rhe lonv.i tudinal strains were :found to vary linearly

with the compressive load.

The specimen A was failed under a post-buckling

load at zero internal pressure a:fter the pressurization

tests were completed (Fip. 10). The post-buckling load

was 12,950 lbs.

33

Fig. 1 0 . Post-buckling railure or specimen A.

34

v

CONCLUSIONS

From the theoretical and experimental results shown

in Fig. 11, the internal pressure is seen to have an

appreciable strenpthening errect on the cylindrical

shell. In FiP-. 11 the increment or buckling stress ~~cr

due to the presence or internal pressure is plotted

a~ainst the internal pressure. The increment or buckling

stress 6~cr is the dirrerence between the buckling stress

with the pressure ~cr and that without the pressure

The theory gives a :fairly good prediction

or the increase o:f compressive buckling stress that may

be expected as a result o:f internal pressure. The curves

for the specimens indicated the common trend o:f an

increase in bucklinF stress for an increase in internal

pressure. The discrepancies between the theoretical

curve and the experimental curves o:r Fig. 11 is believed

to be caused by such factors as manu:racturing

imper:fections in the specimens, end condition ef:fects,

length o:f specimens, and material irregularities, which

were not included in the theory.

The results o:f the strains in the circum:ferential

direction are mainly a :function o:f internal pressure and

do not depend primarily upon the compressive load.

0

" IP. ....-... ~

.~ ..__,

~

.~ II

~

tb0

<1

• 20

.15

.10

.05

"W Theoretical

® Specimen A

0 Specimen B

p = ~H~t Fig. 11. Comparison or theoretical and experimental

re~ult~ showing the increment or buckling ~tre~~ due to internal pre~sure.

35

36

VI

R ECO MN.h.NDATI 0 N S

An extensive experimental program should be

conducted under· relatively constant conditions. Tbis is

reauired in order to describe more precisely the effects

of radius-thickness ratio and length-radius ratio.

Curves or ~cr plotted a~ainst L/R for various R/t ratios

would illustrate these relationships. The experimental

program should involve many identical specimens to give

a reasonable statistical sample ror each combination of

values or L/R and R/t. Care should also be taken to

eliminate the possibilities or yielding at the cylinder

ends, or over-all plastic buckling; and separate

determinations of these effects should be made with many

different materials.

Since the experimental data was limited to values of

~ less than 0.09, additional experimental data is needed

to check the theoretical values for ' greater than 0.09

and beyond the cut-orr point of ~ = 0.168.

Future theoretical work should include a better

larpe-deflection analysis, includinv the errects of

finite len~th, end conditions, and plasticity.

BIBLIOGRAPHY

1. TIMOSHENKO, S.P. (1956) Strength o~ materials, Part II. D. Van Nostrand Co., Inc. Princeton, N.J.

2. LUR'E, A.I. (1947) Statics o~ thin-walled shells.

37

State Publishing House or Technical and Theoretical Literature. Moscow-Leningrad. Translation series or the United States Atomic Energy Commission, AEC-tr-3798.

3. GOODIER, J.N. and HOFF, N.J. (1960) Structural mechanics, proceedings or the rirst symposium on Naval structural mechanics. Pergamon Press, Inc. New York, N.Y. p. 115-168.

4. GOL 1 DE~EIZER, A.L. (1961) Theory o~ elastic thin shells. Pergamon Press, Inc. New York, N.Y.

5. KOlTER, W.T. (1960) Proceedings or the symposium on the theory o~ thin elastic shells. International Union or Theoretical and Applied Mechanics, North-Holland Publishing Co. Amsterdam, Netherlands. p. 167-188.

6. PFLUGER, A. (1961) Elementary statics or shells. F.W. Dodge Corporation. New York, N.Y.

7. NOVOZHILOV, V.V. (1959) The theory o~ thin shells. Translated rrom Russian by P.G. Lowe. P. Noordho~f Ltd. Groningen, Netherlands.

8. VON KARMAN, T. and TSIEN, H.S. (1941) The buckling or thin cylindrical shells under axial compression. Journal or Aeronaut. Sciences 8. p. 303-312.

9. DONNELL, L.H. and WAN, C.C. (1950) Effect of imperfections on buckling or thin cylinders and columns under axial compression. Journal or Applied Mechanics 17. p. 73-88.

10. DONNELL, L.H. (1936) A new theory for buckling of thin cylinders under axial compression and bending. Trans. Amer. Soc. Mech. Engrs. 56. p. 795-806.

11. LO, H., CRATE, H., and SCHWARTZ, E.B. (1950) Buckling of thin-walled cylinder under axial compression and internal pressure. NACA Technical Note 2021. Waehington.

38

12. F'UNG, Y.C. and SECHLER, E.E. (19.57) Buckling of thin-walled circular cylinders under axial compreseion and internal pressure. Journal of Aeronaut. Sciences 24. p. 351-3.56.

13. HARRIS, L.A., SUER, H.S., SKENE, W.T., and BENJAMIN, R.J. (19.57) The stability of thin-walled unstiffened circular cylindere under axial compression including the effects of internal pressure. Journal of Aeronaut. Sciences 24. p. 587-.596.

14. WILSON, W.M. and NEWMARK, N.M. (1933) The strength of thin cylindrical shells ae columns. En~ineering Experiment Station, Univ. of Illinois. Bull. 2.5.

1.5. SERGE.'V, S. and WAL'rON, D.A. (1962) Strenp:th or preesurized columns. The Enpineering Experiment Station, Univ. of Washinpton. Vol. 1!~, No. 2.

16. SERGEV, S. (1961) On column behavior. Journal of Aerospace Sciencee 28. p. 348-349·

17. BROWN, J.K., REA, R.H., and BRh~ER, D.W. (1961) An experimental study or buckling of thin­walled, pressurized, conical shells under compression and compression-bending interaction. Journal or Aerospace Sciences 28. p. 506-.507.

18. LEE, L.H.N. (1962) Inelastic buckling of initially imperfect cylindrical Rhells subject to axial compres~ion. Journal of Aerospace Sciences 29. p. 87-9.5.

19. ZENDER, G.W. (1962) The bending strength of pressurized cylinders. Journal of Aerospace Sciences 29. p. 362-363.

20. WEINGARTEN, V.I. (1962) Effects of internal pressure on the bucklinp of circular-cylindrical shell~ under bending. Journal of Aerospace Sciences 29. p. 804-807.

39

21. WEINGARrEN, V.I., NORGAN, E.J., and SEIDE, P. (1965) Elastic stability of thin-walled cylindrical and conical shells under axial compression. A.I.A.A. Journal 3. p. 500-505.

22. TENNYSON, R.C. (1964) Buckling of circular cylindrical shells in axial compression. A.I.A.A. Journal 2. p. 1351-1353·

23. FLUGGE, W. (1932) Die stabilitat der Krelszylinderschale. Ingenieur Archiv 3. Heft 5. p. 436-506.

24. ALMROTH, B.O. and BRUSH, D.O. (1963) Postbuckling behavior or pressure- or core-stabilized cylinders under axial compression. A.I.A.A. Journa.l 1. p. 2338-2341.

25. SEIDE, P. (1962) The stability under axial compression and lateral pressure of circular-cylindrical ahells with a soft elastic core. Journal of Aerospace Sciences 29. p. 851-862.

26. DOW, M.B. and PETERSON, J.P. (1960) Bendinp and compression tests of pressurized ring­stiffened cylinders. NASA TN D-360. Washington.

21. CHU, C.P. (1964) Effect of internal pressure on compressive buckling stress of a cylindrical shell under eccentric loading. M.S. Thesis, Univ. of Mo. at Rolla, Mo.

28. HARRIS, L.A., SUER, H.S., and SKENE, W.T. (1961) Model investigations of unstiffened and stiffened circular shells. Journal of the Soc. ror Experimental Stress Analysis 1. p. 1-9.

29. ALMROTH, B.O., HOLMES, A.M.C., and BRUSH, D.O. (1964) An experimental study of the buckling of cylinder~ under axial compre8sion. Journal or the Soc. ror Experimental Stress Analysi~ 4. p. 263-270.

40

VITA

The author, LeRoy Earl Thompson, son of Nr. and ~1rs.

Henry H. Thompson, was born on May 22, 193u, in Lithium,

Missouri. He received his primary education in McBride,

Missouri and his secondary education at the Perryville

Public High, .Perryville, Nissouri. He received a

Bachelor of Science de~ree in Civil Engineering from the

School of Mines and Metallurpy of the University of

Missouri, Rolla, Missouri in May 1956.

He held the position of an associate structural

engineer with the McDonnell Aircraft Corporation,

St. Louis, Missouri, for the periods from June 1956 to

April 1957, and October 1957 to September 1960. During

the period of April 1957 to October 1957 he was on a leave

or absence from the McDonnell Aircraft Corporation while

serving a tour of military service with the U.S. Army

Corps of Engineers. He has held the position or

Instructor in Civil Engineering at the School of Mines

and Metallurgy of the University of Missouri, Rolla,

Missouri since September 1960.

He has been enrolled in the Graduate School of the

School of Mines and Metallurgy of the University of

Missouri, Rolla, Missouri on a part-time basis since

September 1960.

He was married to Freida Joanne Medley in February

1958, and their children are Julie Kay, Karen Lynne, and

Michael Lee.